Can a court force a father who is an illegal alien to relinquish his parental rights?

In Texas, how common is it for a judge to grant custody and or visitation rights to a father, who is an illegal alien, has never supported the child in any way, who lives thousands of miles away, has never met the child, and has threatened to take the child away to another country if it ever came to that? I am a U.S. citizen and the mother of a 2 and half year old little girl. Her father is an illegal alien and has never provided for her in any way. He only knows of her because I told him,proved to him that he was her father by paying for the DNA test, and I keep him updated on how she is doing, plus I allow him to speak to her. He promised he would relinquish his parental rights if I got married 2 years ago. Now that I am asking him for just that, he won't. Even though I don't plan to cut him off from her. I promised him I would continue to inform him of her and allow him to talk to her. Still he says no. I am now afraid that he is going to sue me for custody. Can he and what are the chances of him winning? I know that I can get a judge to relinquish his rights but I am very afraid that I will loose. I already have a meeting with a lawyer but not till next week. Does he have a chance of custody and do I have a chance in relinquishing his rights? Any input will help, thank you.